Distributions Frames and bases

Abstract

In this paper we will consider, in the abstract setting of rigged Hilbert spaces, distribution valued functions and we will investigate, in particular, conditions for them to constitute a "continuous basis" for the smallest space $\mathcal{D}$ of a rigged Hilbert space. This analysis requires suitable extensions of familiar notions as those of frame, Riesz basis and orthonormal basis. A motivation for this study comes from the Gel'fand-Maurin theorem which states, under certain conditions, the existence of a family of generalized eigenvectors of an essentially self-adjoint operator on a domain $\mathcal{D}$ which acts like an orthonormal basis of the Hilbert space $\mathcal{H}$. The corresponding object will be called here a {\em Gel'fand distribution basis}. The main results are obtained in terms of properties of a conveniently defined {\em synthesis operator}.

28 pages

References in corpus (5)

• Operator (quasi-)similarity, quasi-Hermitian operators and all that
• Multipliers for Continuous Frames in Hilbert Spaces
• Non-self-adjoint hamiltonians defined by Riesz bases
• Reproducing pairs and the continuous nonstationary Gabor transform on LCA groups
• Reproducing pairs and Gabor systems at critical density

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